The result for the index of \((E^\bullet _2,\eth _2)\)  matches with [1]}, upon using (REF ), while the complex \((E^\bullet _3,\eth _3)\)  cannot be obtained in the five-dimensional framework from a Sasaki manifold but is a new resultIn \(d=5\)  one usually considers only (contact) instantons. Along the lines of [2]} reducing along different fibers produces different distributions of instantons/anti-instantons in \(d=4\) . \((E^\bullet _1,\eth _1)\)  and \((E^\bullet _2,\eth _2)\)  can be obtained in this way, however this is not the case for \((E^\bullet _3,\eth _3)\) . To find such complex one would have to start from \(d=7\) .. Finally, also in this case \(_{\mathrm {eq},H}(E^\bullet _2,\eth _2)\)  and \(_{\mathrm {eq},H}(E^\bullet _3,\eth _3)\)  are symmetric under a reflection \((n_1,n_2)\rightarrow (-n_1,-n_2)\)  hence the final results are independent of the initial choice of deformation map.
